Stellar Astrophysics: Heat Transfer 1. Heat Transfer

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1 Stellar Astrophysics: Heat Transfer 1 Heat Transfer Update date: September 3, 21 We discuss various energy transfer mechanisms in stars: radiative, conductive, and convective. 1 Radiative Transfer 1.1 The Diffusion Equation Central to the discussion of radiative transfer is the specific intensity, I(θ), which is the energy flux per steradian, in the direction of θ. We define the θ = to be the outward z-direction (under the plane-parallel 1-D approximation). The net change in I(θ) along a path ds with the mass emissivity (j) and opacity (κ) is Define the optical depth for an outward-directed ray as di(θ) = [j κi(θ)]ρds (1) τ(z) z z κρdz (2) where z corresponds to the the true surface of the the star where density is approximately zero, hence τ(z ) =. Notice that dz = µds, the above equation of transfer becomes di(τ, µ) µ dτ = I(τ, µ) S(τ, µ) (3) where the source function S(τ, µ) = j/κ. When scattering is important and is isotropic, j should be replaced by J + κ sca µ, where J = 1 4π IdΩ is the mean intensity of the radiation field. Sτ(z) z z κρdz (4) To solve the above equation, we multiply both sides with e τ/µ and then get d [ ] e τ/µ I = e τ/µ S dτ µ (5) Then we have I(τ, µ) = e (τ τ)/µ I(τ, µ) + τ τ (t τ)/µ S(t, µ) e dt (6) µ

2 Stellar Astrophysics: Heat Transfer 2 Depending on the range of µ, we choose different values for τ in simplifying the solution: an outward direction, choosing τ (i.e., the reference level lies deep within the star), I(τ, µ ) = τ (t τ)/µ S(t, µ) e dt (7) µ For an inward direction, choosing τ = (the true surface where I(τ, µ < ) = ), I(τ, µ < ) = τ (t τ)/µ S(t, µ) e dt. (8) µ Of course, the above quantities are all energy dependent (i.e., we have left out the subscript, ν). If the deep interior is nearly in LTE, we expect S to be almost isotropic. Hence its Planck function B can be used. It is then reasonable to expand S in a Taylor series in τ which, to first order, is ( B ) S(t) = B(τ) + (t τ). (9) τ τ Inserting this into the above equation, we have ( B ) I(τ, µ ) = B(τ) + µ. (1) τ τ The equation also holds for I(τ, µ < ) for large τ. Keeping only this linear dependence on µ is called the Eddington approximation. The total outward flux is F = 4π 1 I(θ)cosθdΩ = 2π I(µ)µdµ (11) 1 where the azimuthal symmetry around the z-direction is assumed and µ = cosθ. Note that if I is a constant, then F is zero. Therefore, a net flux, or a radiative transfer, requires that I(θ) must be anisotropic. Using the above I(τ) expression and including the subscript ν, we get The total flux is F ν (τ ν ) = 4π 3 F (r) = 4π 3 B ν τ ν = 4π 3 1 dt ρκ dr where the Rosseland mean opacity, κ, is defined by 1 dt B ν ρ ν κ ν dr T (12) B ν dν (13) T 1 [ κ = 1 κ ν B ν T dν ][ B ν T dν ] 1 (14)

3 Stellar Astrophysics: Heat Transfer 3 Because we then have B ν T dν = T F (r) = c dat 4 3ρκ dr This version of F is in the Fick s law form introduced in Chapter 1. B ν (T )dν = c du 4π dt, (15) = c du 3ρκ dr. (16) 1.2 Radiative Opacity Sources Have a look of Fig. 4.2 in HKT for a sample dependence of the opacity as a function of temperature and pressure. Here we have a brief discussion of various processes that are responsible for this dependence. Electron scattering: κ = σn e ρ In the non-relativistic case (T K), the Thompson cross section can be used. Because n e = ρn A (1 + X)/2 for a typical stellar material Free-free absorption: (17) κ e =.2(1 + X) cm 2 g 1 (18) This process is the inverse of normal bremsstrahlung wherein an electron passing by and interacting with an ion emits a photon. κ ff = j brem /B(T ) = 4(X + Y )(1 + X)ρT cm 2 g 1 (19) The above opacity expressions assume that hydrogen is completely ionized. When the temperature drops to below 1 4, other opacity must be considered. Bound-free and bound-bound absorption: κ bf = (Z/.2)(1 + X)ρ(T/5K) 3.5 cm 2 g 1 (2) Bound-bound opacity is more complicated and is typically much less than κ ff and κ bf. H Opacity and Others: In cooler stars, the free-free and bound-free transitions in the negative hydrogen ion, H, becomes important the most important opacity source for the solar atmosphere. Because of the large polarizability of the neutral hydrogen atom, it is possible to attach an extra electron to it with an ionization potential of.75 ev. The formation of H

4 Stellar Astrophysics: Heat Transfer 4 requires the presence of both the neutral hydrogens (i.e., not too high temperature) and electrons (from ionized hydrogen and/or metals); e.g., κ H = (Z/.2)ρ 1/2 (T/5K) 9 cm 2 g 1 (21) for 3 T 6 K, for example (see the textbook). At even lower temperatures, opacity due to the presence of molecules or small grains becomes important. In general, the opacity is a combination of the absorption and scattering: κ = κ abs + κ sca. But photons may be scattered into the beam from other direction: di/ds = ρκ sca φ. If the scattering is isotropic, then φ = 1 IdΩ J, (22) 4π where J is the mean intensity: the scattering redistributes the energy over all angles. There is an intimate relationship between the specific opacity κ abs ν and emission j ν (ν is added back again), which is related to the radiative equilibrium. A condition of steadystate is that the gas not gain or lose energy to the radiation, which requires the balance the emission, j ν dνdω, and absorption, κ abs ν I ν dνdω, (scattering doesn t transfer energy between the radiation and the gas), or where we have assumed j ν is isotropic. ( jν κ abs ν J ν ) dν, (23) Under the LTE, the level populations of the matter are in thermal equilibrium and is completely determined b a temperature T. This equilibrium means the detailed balance must hold, so that j ν = κ abs ν B ν (T ), (24) where B ν (T ) is the Plank function. If the radiation field is, in addition, described by the Plank function, we would have complete thermodynamic equilibrium. 2 Stellar Atmosphere We can construct a simple model for stellar atmospheres, which provides a link to actual light observed from stars. Such a model also provides a stellar surface boundary condition for the stellar structure modeling. This boundary condition is much more realistic than just setting everything to zero at the surface. To the end, we will have defined the effective temperature and density at the so-called stellar photosphere.

5 Stellar Astrophysics: Heat Transfer A Grey Atmosphere model This simple model will provide the temperature and density as a function of the optical depth. The derivation for the temperature distribution is elegant, without an extra assumption for the isotropy of the outgoing radiation intensity at all outgoing angles but zero for all inward radiation, as required in HKT. Starting from the radiative transfer equation µ di ν dτ ν = I ν S ν, (25) where S ν 1 κ ν (j ν + κ sca ν J ν ) (26) is the source function. In the LTE, we can write S ν = (1 A ν )B ν + A ν J ν, (27) where A ν κ sca ν /κ ν is the albedo. Assuming that the atmosphere is grey (i.e., κ is independent of ν). Eq. 25 can then be integrated over all frequencies to get Further integrating over all angles gives µ di dτ = I S. (28) 1 df 4π dτ = J S = (1 A)(J B) = (29) The fact that the right-hand side vanishes holds even if A 1, because the condition of the detailed balance (Eq. 24), factoring out the constant κ abs, and integrating over ν. So F is constant, as expected in an atmosphere. Note that J = B does not necessarily imply that I ν = B ν! Now we can integrate the above Fick s law, which can be written as and obtain F = c du 3 dτ, (3) U = 3 c F (τ + τ ), (31) where τ is a constant of integration. Because U = 4πJ/c and J = S, we can write Eq. 28 as µ di dτ = I 3 4π F (τ + τ ). (32)

6 Stellar Astrophysics: Heat Transfer 6 This differential equation can be solved, I(µ, τ = ) = 3 4π F (τ + τ )e τ/µ dτ = 3 4π F (µ + τ ). (33) Now at τ =, all of the flux must be outward-directed (µ > ), so I(µ <, τ = ) = if the star is not irradiated by another source. To determine τ, multiple the above equation with µ and integrate it over all angles to find F = F ( τ ). (34) We therefore have τ = 2/3. Now using P = at 4 /3 and a = 4σ/c as well as defining an effective temperature by the relation F = σteff 4, we have Thus the photosphere, where T (τ p ) = T eff, lies at τ p = 2/3. T 4 (τ) = 1 2 T eff( τ) (35) 2 To find the pressure, one needs to solve the hydrostatic equation dp dτ = g s τ (36) where g = GM/R 2 is a constant at the surface. Thus P (τ) = g s τ dτ κ + P (τ = ) (37) To make matter simple, consider the case where κ is constant (as a version of the grey atmosphere). The pressure at the photosphere can be expressed as P (τ) = 2g s 3κ + P (τ = ) (38) where P (τ = ) at the true surface should be dominated by radiation pressure P rad (τ = ). According to Eq in HKT, P rad (τ = ) = 2 L 3c 4πR, (39) 2 (which differs from the direct integration of Eq. 33 by a constant factor of 17/36, however). we have P (τ) = 2g s 3κ (1 + L/L edd), (4)

7 Stellar Astrophysics: Heat Transfer 7 where L edd = 4πcGM is the Eddington limit, above which the radiative forces exceed gravitational forces, i.e., κ Lκ 4πR 2 c > GM R. (41) 2 The opacity in such a case (e.g., the photospheres of very massive and luminous stars) is usually due to electron scattering. With a hydrogen mass fraction of X =.7 and κ e =.34 cm 2 g 1, the limit is ( ) ( ) Ledd (42) MM L Even if the luminosity is only about 1% or so of this limit, then the dynamics of momentum and energy transfer between the radiation field and matter becomes important and should be accounted for, which is very difficult, though. For most stars, however, the Eddington term can be neglected. If the gas is assumed to be composed only of the sum of ideal gas plus radiation, we can then find the density (implicitly) at the photosphere in 1 3 at 4 eff + ρn A µ kt eff = 2 3 g s κ ρ n T s eff. (43) Realistic atmosphere models are constructed for a grid of g s and T eff values, which give the boundary conditions required at the photosphere. To get the spectral distribution, we need to go back to Eq. 7 and set τ = and S ν = B ν (T ). We have I µ (µ) = 1 µ e t/µ B ν [T (t)]dt (44) Note that I µ (µ) depends on angle: rays propagating in a slant will have a lower intensity. As a result, the edge of visible disk of the sun appears darker than the center, a phenomenon known as limb darkening. 2.2 Sample Observed Spectra Stars are classified according to their luminosities and effective temperatures. The O- and B-stars have strong lines due to HeIλ4471 and HeIIλ4541 (Å). The early (e.g., hotter) A-stars show maximum strength in hydrogen Balmer lines. F- and G-stars are week in hydrogen lines, but exhibits metal lines. The cliff near 4,Å is due to the H and K lines of Ca-II, while the depression known as the G-band is primarily due to Fe. The spectra of the cooler K- and M-stars are dominated by metallic lines and molecular bands.

8 Stellar Astrophysics: Heat Transfer Line Profiles and the Curve of Growth Observing spectral lines with high resolution is not always available. What is often done is to measure the equivalent width (EW), defined as ( W ν = 1 I ) ν dν. (45) I You have encountered this in the ISM class. In a simple case of radiation from a background source incident on a foreground slab of an optical depth of τ ν, the EW of a spectral line is then W ν = (1 e τν )dν. (46) While τ is proportional to the column density of the species that is responsible for the line transition, the plot of W ν -column density is called a curve of growth. For a line transition at a stellar atmosphere, the natural broadening is small compared to other broadening factors: e.g., due to stellar rotation, pressure (orbital perturbation due to the encounter with a particle even a neutral one because of the permanent or momentary dipole; Stark/van der Waals effects), and of course the Doppler effect (due to thermal and/or turbulent motion). The line profile is typically a convolution of the Lorentz profile (mostly due to the pressure broadening) and the Doppler effect. The cross-section for a line transition i j is σ ν = e2 mc f ijφ ν (47) where the first term is the classical oscillator cross-section, f ij is the oscillator strength (e.g., f(hα =.64 and f(hβ =.12) and contains the quantum mechanical details, and φ ν is the Lorentz profile γ/4π φ ν = ( ν) 2 + (γ/4π), (48) 2 in which γ is the sum of the damping constants for the two levels involved and ν = ν ν. The integration of this profile gives a value of π. So the total cross-section is The convolution results in where H(a, u) is the well-studied Voigt function H(a = σ tot = πe2 mc f ij (49) σ ν = e2 π mc f ij H(a, u), (5) ν D γ, u = ν ) = a 4π ν D ν D π e y2 dy a 2 + (u y) 2, (51)

9 Stellar Astrophysics: Heat Transfer 9 where ν D = ν ( 2kT mc 2 ) 1/2. The Voigt function has the following basic properties: and for small a, for small u or if large u. H(a, u = ) = 1, (52) H(a, u) e u2 a π H(a, u) a π H(a, u)du = π, (53) 1 dy a 2 + (u y) = 2 e u2 dy e y2 u 2 = a πu 2 To calculate I ν, we need to combine the line opacity with the continuum opacity and solve the equation of transfer. For simplicity, we are going to assume pure absorption in both the continuum and the line. Under this condition, S ν = B ν, the Planck function. For a plane-parallel atmosphere, the equation becomes (54) (55) µ di ν dτ ν = I ν B ν. (56) The solution of this equation for the emergent intensity at τ ν = is The opacity is given by I(µ) = e τν/µ B[T (τ ν)] dτ ν. (57) µ where κ C ν κ ν = κ C ν + κ L ν, (58) is the continuum opacity and κ L ν = κ L φ ν is the line opacity, with κ L = n i ρ e 2 mc f ij(1 e hν/kt ) (59) being the line opacity at the line center ν. The term e hν/kt accounts for the stimulated transition from j i. We can usually ignore the variation with ν in κ C ν over the width of the line. Let us also assume that β ν κ L ν /κ C ν is independent of τ ν. We can then write dτ ν = (1 + β ν )dτ, where dτ = ρκ C ν dz. Finally, we assume that in the line forming region, the temperature does not vary too much, so that we can expand B ν to first order in τ only, B[T (τ ν )] B + B 1 τ, (6)

10 Stellar Astrophysics: Heat Transfer 1 where B and B 1 = ( B ν ) are constant. Inserting these approximations into Eq. 57, multiplying by µ and integrating over outward bound rays gives us the flux, τ 1 [ F ν =2π [B + B 1 τ] exp τ ] µ (1 + β ν) (1 + β ν )dτdµ [ =π B + 2B ] (61) 1. 3(1 + β ν ) Far form the line center, β ν, implying that the continuum flux is [ Fν C = π B + 2B ] 1. (62) 3 Hence the depth of the line is A ν 1 F ν F C ν = A β ν 1 + β ν, (63) where A is the depth of an infinitely opaque (β ν ) line. Now we can computer the equivalent width, W ν = 2B 1 3B + 2B 1 (64) A β ν 1 + β ν dν (65) Let s change variables from ν to u, as in Eq. 51. Since H(a, u) is symmetrical about the line center, we will just integrate over ν >, giving where β = κ L ν /(κ C ν ν D ). W ν = 2A ν D It is useful to understand the behavior of W ν in various limits: First, at small line optical depth (β ν 1), β H(a, u) du, (66) 1 + β H(a, u) W ν πa ν D β [1 β 2...]. (67) Notice that W ν is independent of the Doppler broadening, since β ν 1 D. Physically, in the limit of small optical depth, each atom in state i sees essentially the same photon intensity, and the flux removed is just proportional to the number of atoms n i.

11 Stellar Astrophysics: Heat Transfer 11 At the other extreme when β ν 1, W ν πaβ A ν D. (68) Notice that aβ ν 2 D. W ν is again independent of the Doppler broadening. In the intermediate case, W ν 2A ν D lnβ. (69) 3 Heat Transfer by Conduction Energy transport by conduction results from collision between electrons and atomic nuclei and works in the same fashion as radiative transport. We can rewrite Fick s law of particle diffusion (Chapter 1) F cond = vλ du 3 dr, (7) into F cond = D e dt dr where the diffusion coefficient, D e, has the general form D e = cvveλ, in which the specific 3 heat at constant volume c V = ( ) U, v T e is the mean velocity of electrons, while the mean free v path λ = 1 σ C n I, where σ C is the Coulomb scattering cross section and n I is the ion number density. We can formally define the conductive opacity as (71) κ cond 4acT 3 3D e ρ. (72) We then have The combined heat flux is F cond = 4acT 3 dt 3κ cond ρ dr F tot = F rad + F cond = 4acT 3 3κ total ρ dt dr (73) (74) where 1 κ total = 1 κ rad + 1 κ cond (75) Note that whichever opacity is the smaller of κ rad or κ cond, it is also more important in determining the total opacity and hence the heat flow, as in a current flowing through a parallel circuit.

12 Stellar Astrophysics: Heat Transfer 12 In a dense electron degenerate material, where the conductive heat transport can dominate, ( κ cond = (4 1 8 cm 2 g 1 ) µ2 e T ) 2, Zc 2 (76) µ I ρ where Z C is the ion charge. For a typical cool white dwarf with ρ = 1 6 g cm 3, T 1 7 K, and a composition of carbon (µ e = 2, µ I = 12, and Z c = 6), κ cond = cm 2 g 1, compared to κ e =.2 cm 2 g 1, the opacity due to electron scattering. In normal stellar material, κ cond is too large to be important. 4 Heat Transfer by Convection When the temperature gradient in a region of a star becomes too large (e.g., due to a large opacity), the gas may become convectively unstable. But we still don t have an accurate theory to describe stellar convection, which is a global and highly non-linear phenomenon. What is often used in the stellar modeling is the mixing length theory, a local phenomenological model, which describes the convection as being composed of readily identifiable eddies or parcels or bubbles, that can move from regions of high heat content to regions of lower one, or conversely. The parcels rise in the star through some characteristic distance, l, the mixing length before they lose their identify or break up and merge with the surrounding fluid. What happens is that large eddies drive progressively small ones until eventually very tiny eddies are excited, with sizes λ ν ν/u ν, where ν and U ν are the viscosity and Key assumptions (part of the so-called Boussingesq approximation) are made in the theory: The characteristic dimension of a parcel is the same order as l. l is much shorter than the stellar scale height (e.g., λ P ). the pressure is balanced between the parcel and its surrounding. the temperature difference between a parcel interior and exterior is small. 4.1 Criteria for Convection We begin by considering a typical parcel created by some unspecified process (e.g., a fluctuation). This parcel has the interior temperature T, pressure P, and density ρ. Outside the parcel, the corresponding quantities are denoted by T, P, and ρ. Suppose that T > T, then Archimedes principle states that the parcel will experience a net upward buoyancy force of (ρ ρ )gv where g is the local gravity and V is the volume of the

13 Stellar Astrophysics: Heat Transfer 13 parcel. The force will lead the parcel to rise (here in the direction of the z-axis). Assuming that there is no heat exchange with the surrounding, T will vary as the parcel rises: β ad dt ( dlnt dz = T dlnp dlnp )ad dz The corresponding variation of the temperature in the surrounding gas is β dt dz dlnt dlnp = T dlnp dz Because of the pressure balance, we can replace P with P ; because of the small T variation, we can replace the lone T with T. We further define dlnp/dz λ 1 P. Therefore, we have (77) (78) β β ad = T λ P ( ad ) (79) When > ad, the fluid is convectively unstable. The interior temperature of the parcel drops at a rate slower than that of the exterior temperature (i.e., β > β ad ). The parcel will keep rising until a mixing length is traversed. These criteria are called the Schwarzschild criteria. When the convection occurs, however, it must also change the thermal structure and hence,, and so on. So the process is highly non-linear. Also in real life, heat must be exchanged between our parcel and the exterior. But we will see that these do not change the criteria. 4.2 Radiative Leakage To follow what happens in the parcel as it moves, we consider ( ) dt T dt = + w T (8) t P where w is the parcel velocity. The term ( ) T takes care of the instantaneous heat change, t P ( ) T ρc P = F rad = K 2 T, (81) t P where P V work of the parcel is neglected and K = 4acT 3 3κρ. (82) While we have little idea as to what the exact structure of our parcel, we approximate 2 T = T/l 2, where T = T T. Hence, we have T t = ν T T (83) l2

14 Stellar Astrophysics: Heat Transfer 14 where ν T = K/ρc P. The remaining r.h.s. advective (adiabatic) term of Eq. 8 w T = w T = β z ad describes how T behaves without the radiative loss. The corresponding change of the ambient temperature as seen by the moving parcel is Putting all these together, we have d T dt dt dt = dt dz dz dt = βw (84) = ν T l 2 T + (β β ad)w. (85) This equation describe the time-dependent temperature contrast between the parcel and its immediate surroundings as the parcel moves. 4.3 Equation of Motion The motion caused by the buoyancy force is approximately where we have ignored any viscous effects. For a small relative density contrast, (ρ ρ ) ρ dw dt = (ρ ρ ) g (86) ρ = Q (T T ) T where Q is the coefficient of thermal expansion, defined as ( dlnρ ) Q dlnt = Q T (87) T Thus, dw dt = Qg T (89) T which is to be considered along with the above energy equation. (88) 4.4 Convective Efficiencies If we suppose that all the coefficients in Eqs. 85 and 89 are constant, not only over the distance l but for all time, then we can easily solve for T and w as functions of time. The solutions have the form e σt, where σ is a complex angular frequency.

15 Stellar Astrophysics: Heat Transfer 15 It is easy to find that σ must satisfy the characteristic equation where the Brunt-Väisälä frequency, N, is defined by σ 2 + σ ν T l 2 + N 2 = (9) N 2 = Qg T (β β ad). (91) If N 2 <, σ is real with one positive root; that is, both T and w may increase exponentially the fluid is convectively unstable. This is equivalent to (β β ad ) > ), as derived in the case of no heat exchange. What limits the motion in the context of the MLT is the ultimate breakup of the parcel. If ν T /l 2 N, then σ = N. This is the case of efficient convection because the parcel loses essentially no heat during its travel until it breaks up. Conversely, σ = N 2 l 2 /ν T 1, so T and w increase, but slowly. From now on, our analysis will assume ν T, or the case of adiabatic convection. If N 2 > (no convection), σ has an imaginary part and the parcel oscillates, which will be damped out by the negative real part due to radiative losses. The magnitude of w may be estimated as Substitute w, we have d T dt w = σl (92) = σ T = (β β ad )w ν T T (93) l2 T = σl(β β ad) σ + ν = σ2 lt T l 2 Qg, (94) where the last step is obtained from Eq. 89. Convection fluxes F conv = ρwc P T = ρσ3 l 2 c P T Qg For adiabatic convection, σ = N 2, using Eq. 79, one gets. (95) F conv = ρc P l 2 T (gq) 1/2 ( ad ) 3/2. (96) When the convection efficient (adiabatic), σ = N 2 ( ad ) 1/2 There is no exact way to deal with l. It may be chosen to be a fraction (α called the mixing length parameter) of the λ P (e.g.,.5) and adjusted from comparison of obtained evolutionary models with observations. This treatment is often not self-consistent. But fortunately, it does not much a difference if the convection is efficient. λ 3/2 P

16 Stellar Astrophysics: Heat Transfer Model convection implementation While thus far we have assumed that (or β) and l are known so that we can compute σ etc., and finally, F conv. But there is no guarantee that the system is consistent, since the onset of the convection can substantially change the stellar structure. Thus some sort of iteration is then required. Furthermore, the total flux in general is compounded from the convection (if present) and the radiative flux (which is always there). How is this situation handled? For simplicity, we consider here only adiabatic convection. To see whether or not the convection may play a role in the heat transfer, we first make believe that all the heat flux, which often depends little on the local heat transfer (especially in a stellar envelope), and compute rad = 3 4ac r 2 P κ T 4 F tot GM r. (97) This rad, derived from the global quantity (F tot ) is more reliable than the local (from the assumed heat transfer in the present modeling). So we need to check the consistency. If = rad, then the flux is carried solely by the radiation and you are done. However, if > rad, the convection must then play a role with F tot = F rad + F conv, (98) where F rad = ρc pt ν T, (99) λ P where F rad = K T is used. Then we can now solve for and compare it with the present model value. We may need to integrate to make them converge. When the convection is efficient, it dominates. It is often reasonable just to assume = ad. 5 Review Key concepts: Radiation transfer, Rosseland mean opacity, Grey atmosphere, curve of growth, heat conduction, efficient convection, adiabatic convection. What are the various heat transfer mechanisms? What are the physical conditions for each of the mechanisms to dominate? What the various physical processes that contribute to the stellar opacity (as a function of temperature in a plot)? What are the power law exponents of these processes when the opacity is expressed in a power law? Why do they have these particular exponents? What are the relative importance of the processes in various parts from the stellar center to surface?

17 Stellar Astrophysics: Heat Transfer 17 How do the processes depend on metallicity or He abundance? What is the Eddington limit? Please derive the Schwarzschild criteria for the convective instability by yourself. What are the underlying assumptions? What is the mixing length theory? Why do main-sequence massive stars tend to have convective cores and radiative envelops, whereas lower mass stars tend to have the opposite behaviors? What is the practical way in the stellar modeling to decide on whether or not the convection is important? How may it be implemented?